If during the fermentation of sugar solution that is `0.12` M the concentration of sugar is reduced to `0.06` M in 10 h and to `0.03` M in 20 h the order of reaction is

Options

A. 1
B. `3//2`
C. 2
D. None of these

Explanation

To determine the order of the reaction based on the concentration changes over time, we can use the data provided about the sugar concentration during fermentation.

### Step-by-Step Solution:

1. **Identify Initial and Final Concentrations:**
- Initial concentration of sugar, \( [A_0] = 0.12 \, M \)
- Concentration after 10 hours, \( [A] = 0.06 \, M \)
- Concentration after 20 hours, \( [A] = 0.03 \, M \)

2. **Calculate the Change in Concentration:**
- Change from 0.12 M to 0.06 M in 10 hours:
\[ \Delta [A]_{10h} = [A_0] - [A] = 0.12 \, M - 0.06 \, M = 0.06 \, M \]
- Change from 0.06 M to 0.03 M in the next 10 hours:
\[ \Delta [A]_{20h} = [A]_{10h} - [A]_{20h} = 0.06 \, M - 0.03 \, M = 0.03 \, M \]

3. **Determine the Rate of Reaction:**
- The average rate of reaction for the first 10 hours:
\[ \text{Rate}_{10h} = \frac{\Delta [A]_{10h}}{\Delta t} = \frac{0.06 \, M}{10 \, h} = 0.006 \, M/h \]
- The average rate of reaction for the next 10 hours:
\[ \text{Rate}_{20h} = \frac{\Delta [A]_{20h}}{\Delta t} = \frac{0.03 \, M}{10 \, h} = 0.003 \, M/h \]

4. **Analyze the Rates:**
- The rate of reaction decreases as the concentration of sugar decreases. This suggests that the reaction is not zero-order (where the rate would remain constant regardless of concentration).

5. **Use the Integrated Rate Law:**
- For a first-order reaction, the relationship is:
\[ \ln \left( \frac{[A_0]}{[A]} \right) = kt \]
- For a second-order reaction, the relationship is:
\[ \frac{1}{[A]} - \frac{1}{[A_0]} = kt \]

6. **Check the Order of Reaction:**
- Calculate for first-order:
\[ \ln \left( \frac{0.12}{0.06} \right) = \ln(2) \approx 0.693 \quad \text{(for 10 hours)} \]
\[ \ln \left( \frac{0.06}{0.03} \right) = \ln(2) \approx 0.693 \quad \text{(for 20 hours)} \]
- Since the logarithmic change is consistent, it indicates a first-order reaction.

7. **Conclusion:**
- The order of the reaction is **first-order**.